Geometry of Moduli Spaces of Complex Curves

Abstract

In this chapter we present an overview of the connection between the geometry of moduli spaces of complex curves with marked points and the topology of embedded graphs. According to Harer [137], the idea of this connection belongs to Mumford. It proved to be extremely fruitful. The most celebrated results here are the calculation of the orbifold Euler characteristic of moduli spaces of smooth curves due to Harer and Zagier [138] (we present also Kontsevich’s calculation based on similar ideas) and Kontsevich’s proof of Witten’s conjecture. We must note that a complete exposition of this proof containing all the details has not yet been published. Our text does not fulfill this mission either. In the next chapter we will show how the geometry of moduli spaces of curves is related to that of Hurwitz spaces, that is, the moduli spaces of meromorphic functions on complex curves. Since a meromorphic function is also associated to an embedded graph, we obtain another facet of the same connection.KeywordsModulus SpaceLine BundleIrreducible ComponentMarked PointEuler CharacteristicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.